Representations by certain octonary quadratic forms with coefficients 1, 5, or 25

Abstract

It is an important objective to determine the number of representations of a positive integer by certain quadratic forms in number theory. Formulae for $% N(1^{2i},2^{2j},3^{2k},6^{2l};n)$ for the nine octonary quadratic forms appear in the literature, whose coefficients are $1,2,3$ and $6$. Moreover, the formulae for $N(1^{i},3^{j},9^{k};n)$ for several octonary quadratic forms have been given by Alaca. Here, we determine formulae, for $N(1^{i},5^{j},25^{k};n)$ for several octonary quadratic forms.